Are the can lights and receptacles all on the same 20A circuit from the panel?

Do you see the same 15.6V at the recptacles with the lights on and off?

If the current-carrying conductors to the lights and the receptacles run alongside each other for any distance, I'm suspecting a small induced voltage on the hot lead to the receptacles coming from the current flowing to power the lights. With no load on the receptacles and a sensitive meter, you will see that voltage. Try plugging in a small lamp or other load and I'll bet the voltage drops to essentially zero.